nLab higher Kac-Moody algebra

Idea

• In dimension $1$, we have that $\mathcal{O}(\mathbb{A}^1-\{0\})\otimes\mathfrak{g} = \mathfrak{g}[z,z^{-1}]$ and, thus, we can consider its non-trivial central extensions by $2$-cocycles. These are the essential ingredients of an (ordinary) Kac-Moody algebra.

• In higher dimension, Hartogs' extension theorem tells us that $\mathcal{O}(\mathbb{A}^n-\{0\})\otimes\mathfrak{g} \cong \mathcal{O}(\mathbb{A}^n)\otimes\mathfrak{g}$, which means that we do not have any interesting central extension. On the other hand, the punctured affine spaces $\mathbb{A}^n-\{0\}$ have non-trivial higher cohomology.

The idea underpinning the definition of higher Kac-Moody algebras in FHK 19 is that the natural framework to solve this issue and construct a higher-dimensional generalization of Kac-Moody algebras is derived algebraic geometry. This is achieved by replacing the algebra $\mathcal{O}(\mathbb{A}^n-\{0\})$ with the dg-algebra $\mathbb{R}\Gamma(\mathbb{A}^{n}-\{0\},\mathcal{O})$ of derived sections, which allows interesting central extensions.

This principle can be applied not only to punctured affine spaces $\mathbb{A}^{n}-\{0\}$, but also to punctured formal disks $\mathbb{D}_n^\circ$.

Details

Faonte-Hennion-Kapranov higher Kac-Moody algebras

Let $X$ be an $n$-dimensional variety over $\mathbb{C}$ and $\mathfrak{g}$ a Lie algebra. We can define the higher current algebra by the dg-Lie algebra

$$\mathfrak{g}_n \;=\; \mathfrak{g}\otimes\mathbb{R}\Gamma(\mathbb{D}_n^\circ,\mathcal{O}_X),$$

where $\mathbb{R}\Gamma(\mathbb{D}_n^\circ,\mathcal{O}_X)$ is the commutative dg-algebra of derived global sections of the structure sheaf $\mathcal{O}_X$ on the punctured formal disk $\mathbb{D}_n^\circ = \mathrm{Spec}(\mathbb{C}[[z_1,\dots,z_n]])-\{0\}$.

A higher Kac-Moody algebra $\widehat{\mathfrak{g}}_{n,\Theta}$ is the central extension of the higher current algebra $\mathfrak{g}_n$ by an invariant polynomial $\Theta$ on $\mathfrak{g}_n$ of degree $(n+1)$.

Gwilliam-Williams higher Kac-Moody factorisation algebras

Let $X$ be a complex manifold of dimension $n$ equipped with a holomorphic principal bundle $P\rightarrow X$ with structure group $G$.

Let $\mathfrak{ad}(P) \coloneqq P\times_G\mathfrak{g}$ be the adjoint bundle associated to $P$. Now, let $\mathscr{Ad}(P)$ be the local Lie algebra whose sections are $\Omega_c^{0,\ast}(U,\mathfrak{ad}(P))$ on any open set $U\subset X$ and whose differential is the $(0,1)$-connection $\bar{\partial}_P$.

Let $\Theta$ be a degree $1$ cocycle in the local Chevalley-Eilenberg cochains $\mathrm{CE}_{\mathrm{loc}}(\mathscr{Ad}(P))$, which defines a $1$-shifted central extension $\widehat{\mathscr{Ad}(P)}_\Theta$.

The higher Kac-Moody factorization algebra on $X$ of type $\Theta$ is defined in GW 21 as the twisted enveloping factorization algebra $\mathbb{U}_\Theta(\mathscr{Ad}(P))$ whose sections are

$$\mathbb{U}_\Theta\left(\mathscr{Ad}(P)\right)(U) \;=\; \Big(\mathrm{Sym}\left(\Omega_c^{0,\ast}(U,\mathfrak{ad}(P))[1]\right), \; \bar{\partial} + \mathrm{d}_\mathrm{CE} + \Theta \Big)$$

on any open set $U\subset X$.

Relation between the two

Let $r : \mathbb{A}^{n}_{\mathbb{C}} - \{0\} \rightarrow (0,+\infty)$ be the radial projection map sending $(z_1,\dots,z_n)\mapsto \sqrt{|z_1|^2+\dots+|z_n|^2}$.

In GW 21 the following map of factorization algebras on the positive reals is constructed:

$$\mathbb{U}(\hat{\pi}_{\mathfrak{g},n,\Theta}) \, : \; \mathbb{U}(\Omega^{0,\ast}_c\otimes\hat{\mathfrak{g}}_{n,\Theta}) \; \longrightarrow \; r_\ast\mathbb{U}_\Theta(\mathscr{Ad}(P)),$$

where on the left-hand side we have the enveloping factorization algebra which encodes the enveloping $A_\infty$-algebra of the FHK 19 higher Kac-Moody algebra $\hat{\mathfrak{g}}_{n,\Theta}$.

This map establishes a relation between derived algebraic geometry and quantum field theory, formulated the language of factorization algebras. In particular, the higher Kac-Moody algebra $\hat{\mathfrak{g}}_{n,\Theta}$ “controls” its corresponding higher Kac-Moody factorization algebra just like an affine Kac-Moody algebra “controls” its corresponding vertex algebra.

Examples

Ordinary Kac-Moody algebras

For $n=1$ and $X=\mathbb{A}^1_{\mathbb{C}}$, one recovers the formal current algebra $\mathfrak{g}_1 = \mathfrak{g}((z))$, whose central extensions $\tilde{\mathfrak{g}}_{1,\Theta}$ are ordinary Kac-Moody algebras.

Affine space $\mathbb{A}^n_{\mathbb{C}}$

Consider the affine space $X=\mathbb{A}^n_{\mathbb{C}} = \mathrm{Spec}(\mathbb{C}[z_1,\dots,z_n])$. The cohomology of the complex of derived global sections $\mathbb{R}\Gamma(\mathbb{D}_n^\circ,\mathcal{O}_X)$ will be the following:

$$\mathrm{H}^i(\mathbb{D}_n^\circ,\mathcal{O}_X) \;=\; \begin{cases}\mathbb{C}[[z_1,\dots,z_n]], & i=0\\z_1^{-1}\cdots z_n^{-1}\mathbb{C}[z_1^{-1},\dots,z_n^{-1}], & i=n-1\\ 0, & \text{otherwise}\end{cases}$$

Related concepts

• Kac-Moody algebras

• derived algebraic geometry

• factorization algebras

References

Higher Kac-Moody algebras were proposed in

• G. Faonte, B. Hennion and M. Kapranov, Higher Kac-Moody algebras and moduli spaces of $G$-bundles, Advances in Mathematics, Volume 346, 13 April 2019, pages 389-466, math.AG/1701.01368.

• M. Kapranov, Infinite-dimensional (dg) Lie algebras and factorization algebras in algebraic geometry, Japanese Journal of Mathematics volume 16, pages 49–80 (2021) https://doi.org/10.1007/s11537-020-1921-4.

Higher Kac-Moody algebras casted in the language of factorization algebras:

• Owen Gwilliam, B. R. Williams, Higher Kac-Moody algebras and symmetries of holomorphic field theories, Adv. Theor. Math. Phys. 25 1 (2021) 129-239 [math.QA/1810.06534]

A variant of this construction on the product of a worldsheet and a spectral plane (i.e. another copy of the complex plane $\mathbb{C}$ with marked points which defines any rational Gaudin model):

• Luigi Alfonsi, Charles A. S. Young, Higher current algebras, homotopy Manin triples, and a rectilinear adelic complex, Journal of Geometry and Physics 191 (2023) 104903 [doi:10.1016/j.geomphys.2023.104903, math.QA/2208.06009]

Review:

• Luigi Alfonsi, §7 in: Higher geometry in physics, in: Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2312.07308]